Table 2 Open in new tab Simulations with...

Approach:	R-INLA 1	R-INLA 2	JMbayes2	rstanarm 1	rstanarm 2
$β_{10} = 0.2$	0.001	(0.049)	95	0.001	(0.049)	96	0.002	(0.049)	95	0.006	(0.051)	93	–0.001	(0.048)	96
$β_{11} = - 0.1$	0.003	(0.019)	96	0.003	(0.019)	95	0.002	(0.019)	95	0.003	(0.021)	93	–0.001	(0.019)	94
$β_{12} = 0.1$	0	(0.04)	95	0	(0.04)	94	–0.001	(0.04)	94	–0.004	(0.041)	93	0.004	(0.039)	97
$β_{13} = - 0.2$	–0.001	(0.034)	97	0	(0.034)	98	0	(0.034)	98	–0.003	(0.035)	96	–0.007	(0.033)	98
$σ_{ε 1} = 0.4$	0	(0.003)	94	0	(0.003)	94	0	(0.003)	94	0.002	(0.01)	92	0.001	(0.004)	95
$β_{20} = 3$	–0.002	(0.053)	96	–0.002	(0.053)	95	0.002	(0.053)	96	0.001	(0.049)	98	–0.008	(0.047)	98
$β_{21} = - 0.1$	–0.002	(0.02)	96	–0.002	(0.02)	96	–0.001	(0.02)	96	–0.002	(0.019)	97	–0.003	(0.019)	98
$β_{22} = 0.1$	0.002	(0.043)	96	0.002	(0.043)	95	0	(0.043)	96	0.001	(0.041)	95	0.005	(0.041)	95
$β_{23} = - 0.2$	0.004	(0.045)	94	0.004	(0.045)	94	0.003	(0.045)	95	0.002	(0.045)	94	0.009	(0.04)	96
$β_{30} = 1$	–0.014	(0.092)	95	–0.014	(0.091)	95	–0.005	(0.093)	94	0.001	(0.096)	94	0.026	(0.102)	97
$β_{31} = - 1$	0.008	(0.031)	93	0.009	(0.031)	93	–0.001	(0.031)	95	–0.001	(0.032)	94	–0.007	(0.031)	96
$β_{32} = 1$	–0.006	(0.074)	96	–0.007	(0.073)	95	0.004	(0.075)	97	–0.002	(0.076)	94	–0.008	(0.071)	98
$β_{33} = - 1$	0.01	(0.07)	95	0.011	(0.071)	95	–0.001	(0.071)	95	–0.003	(0.068)	96	–0.018	(0.077)	85
$σ_{b 10}^{2} = 0.16$	0.003	(0.016)	95	0.002	(0.014)	94	–0.001	(0.014)	99	–0.007	(0.029)	94	–0.008	(0.021)	96
$σ_{b 11}^{2} = 0.09$	0.003	(0.01)	94	0.004	(0.009)	95	0	(0.009)	93	0.008	(0.038)	88	0.011	(0.025)	94
$σ_{b 20}^{2} = 0.25$	0.002	(0.019)	95	0.001	(0.017)	96	0.002	(0.017)	95	0.004	(0.017)	96	0.002	(0.016)	95
$σ_{b 21}^{2} = 0.16$	0.002	(0.014)	95	0.002	(0.013)	95	0.003	(0.014)	94	0.002	(0.012)	97	0.007	(0.016)	82
$σ_{b 30}^{2} = 0.25$	–0.01	(0.036)	96	–0.009	(0.037)	96	0	(0.041)	93	0.004	(0.041)	96	0.002	(0.037)	97
${cov}_{b 10, b 11} = 0.03$	–0.002	(0.009)	96	–0.004	(0.009)	94	–0.001	(0.007)	97	–0.001	(0.01)	96	–0.003	(0.009)	96
${cov}_{b 10, b 20} = 0.02$	0	(0.011)	95	0	(0.011)	95	0	(0.01)	96	–0.001	(0.01)	94	–0.003	(0.011)	97
${cov}_{b 10, b 21} = 0.04$	–0.001	(0.01)	96	–0.001	(0.01)	96	0	(0.009)	96	–0.002	(0.011)	94	0	(0.009)	97
${cov}_{b 10, b 30} = 0$	0	(0.019)	96	–0.003	(0.017)	97	0.001	(0.016)	95	–0.002	(0.016)	94	–0.001	(0.017)	95
${cov}_{b 11, b 20} = 0.03$	–0.001	(0.01)	96	–0.001	(0.01)	96	–0.001	(0.009)	96	0	(0.011)	94	–0.002	(0.01)	97
${cov}_{b 11, b 21} = 0$	0	(0.008)	95	0	(0.008)	96	0	(0.008)	95	0.002	(0.01)	92	0	(0.008)	96
${cov}_{b 11, b 30} = - - 0.06$	0.005	(0.014)	92	0.006	(0.013)	94	0.002	(0.014)	93	0	(0.015)	95	–0.001	(0.014)	97
${cov}_{b 20, b 21} = 0.08$	0	(0.012)	95	0	(0.012)	95	–0.001	(0.011)	94	0	(0.011)	96	0.002	(0.011)	96
${cov}_{b 20, b 30} = 0.05$	–0.001	(0.019)	95	–0.002	(0.019)	95	–0.003	(0.018)	94	–0.001	(0.019)	96	0.001	(0.018)	96
${cov}_{b 21, b 30} = 0.04$	–0.001	(0.017)	95	–0.001	(0.017)	96	–0.003	(0.016)	94	0.009	(0.019)	92	0.011	(0.018)	95
$φ_{1} = 0.5$	–0.003	(0.101)	95	–0.003	(0.102)	94	0	(0.103)	98	0.009	(0.113)	94	0.021	(0.114)	96
$φ_{2} = - 0.2$	0.006	(0.074)	93	0.005	(0.073)	94	0.005	(0.074)	97	0.002	(0.077)	93	0.006	(0.072)	94
$φ_{3} = 0.3$	–0.001	(0.094)	94	0.001	(0.092)	94	–0.004	(0.092)	94	–0.006	(0.093)	94	–0.003	(0.087)	95
Conv. rate	1	1	1	0.51	0.64
Comp. time (s)	43.55 (5.58)	69.8 (44.59)	295.73 (20.22)	3766.41 (1429.06)	10,861.22 (3735.02)

Approach:

R-INLA 1

R-INLA 2

JMbayes2

rstanarm 1

rstanarm 2

(Empirical Bayes)

(Full Bayesian)

(1 chain / 1000 iter.)

(4 chains / 2000 iter.)

True value

Bias

(SD)

CP (%)

Bias

(SD)

Bias

(SD)

Bias

(SD)

Bias

(SD)

β_{10} = 0.2

0.001

(0.049)

0.001

(0.049)

0.002

(0.049)

0.006

(0.051)

–0.001

(0.048)

β_{11} = - 0.1

0.003

(0.019)

0.003

(0.019)

0.002

(0.019)

0.003

(0.021)

–0.001

(0.019)

β_{12} = 0.1

(0.04)

–0.001

(0.04)

–0.004

(0.041)

0.004

(0.039)

β_{13} = - 0.2

–0.001

(0.034)

–0.003

(0.035)

–0.007

(0.033)

σ_{ε 1} = 0.4

(0.003)

0.002

(0.01)

0.001

(0.004)

β_{20} = 3

–0.002

(0.053)

–0.002

(0.053)

0.002

(0.053)

0.001

(0.049)

–0.008

(0.047)

β_{21} = - 0.1

–0.002

(0.02)

–0.002

(0.02)

–0.001

(0.02)

–0.002

(0.019)

–0.003

(0.019)

β_{22} = 0.1

0.002

(0.043)

0.002

(0.043)

0.001

(0.041)

0.005

(0.041)

β_{23} = - 0.2

0.004

(0.045)

0.004

(0.045)

0.003

(0.045)

0.002

(0.045)

0.009

(0.04)

β_{30} = 1

–0.014

(0.092)

–0.014

(0.091)

–0.005

(0.093)

0.001

(0.096)

0.026

(0.102)

β_{31} = - 1

0.008

(0.031)

0.009

(0.031)

–0.001

(0.031)

–0.001

(0.032)

–0.007

(0.031)

β_{32} = 1

–0.006

(0.074)

–0.007

(0.073)

0.004

(0.075)

–0.002

(0.076)

–0.008

(0.071)

β_{33} = - 1

0.01

(0.07)

0.011

(0.071)

–0.001

(0.071)

–0.003

(0.068)

–0.018

(0.077)

σ_{b 10}^{2} = 0.16

0.003

(0.016)

0.002

(0.014)

–0.001

(0.014)

–0.007

(0.029)

–0.008

(0.021)

σ_{b 11}^{2} = 0.09

0.003

(0.01)

0.004

(0.009)

0.008

(0.038)

0.011

(0.025)

σ_{b 20}^{2} = 0.25

0.002

(0.019)

0.001

(0.017)

0.002

(0.017)

0.004

(0.017)

0.002

(0.016)

σ_{b 21}^{2} = 0.16

0.002

(0.014)

0.002

(0.013)

0.003

(0.014)

0.002

(0.012)

0.007

(0.016)

σ_{b 30}^{2} = 0.25

–0.01

(0.036)

–0.009

(0.037)

(0.041)

0.004

(0.041)

0.002

(0.037)

{cov}_{b 10, b 11} = 0.03

–0.002

(0.009)

–0.004

(0.009)

–0.001

(0.007)

–0.001

(0.01)

–0.003

(0.009)

{cov}_{b 10, b 20} = 0.02

(0.011)

(0.01)

–0.001

(0.01)

–0.003

(0.011)

{cov}_{b 10, b 21} = 0.04

–0.001

(0.01)

–0.001

(0.01)

(0.009)

–0.002

(0.011)

(0.009)

{cov}_{b 10, b 30} = 0

(0.019)

–0.003

(0.017)

0.001

(0.016)

–0.002

(0.016)

–0.001

(0.017)

{cov}_{b 11, b 20} = 0.03

–0.001

(0.01)

–0.001

(0.01)

–0.001

(0.009)

(0.011)

–0.002

(0.01)

{cov}_{b 11, b 21} = 0

(0.008)

0.002

(0.01)

(0.008)

{cov}_{b 11, b 30} = - - 0.06

0.005

(0.014)

0.006

(0.013)

0.002

(0.014)

(0.015)

–0.001

(0.014)

{cov}_{b 20, b 21} = 0.08

(0.012)

–0.001

(0.011)

0.002

(0.011)

{cov}_{b 20, b 30} = 0.05

–0.001

(0.019)

–0.002

(0.019)

–0.003

(0.018)

–0.001

(0.019)

0.001

(0.018)

{cov}_{b 21, b 30} = 0.04

–0.001

(0.017)

–0.001

(0.017)

–0.003

(0.016)

0.009

(0.019)

0.011

(0.018)

φ_{1} = 0.5

–0.003

(0.101)

–0.003

(0.102)

(0.103)

0.009

(0.113)

0.021

(0.114)

φ_{2} = - 0.2

0.006

(0.074)

0.005

(0.073)

0.005

(0.074)

0.002

(0.077)

0.006

(0.072)

φ_{3} = 0.3

–0.001

(0.094)

0.001

(0.092)

–0.004

(0.092)

–0.006

(0.093)

–0.003

(0.087)

Conv. rate

0.51

0.64

Comp. time (s)

43.55 (5.58)

69.8 (44.59)

295.73 (20.22)

3766.41 (1429.06)

10,861.22 (3735.02)

Approach:	R-INLA 1	R-INLA 2	JMbayes2	rstanarm 1	rstanarm 2
$β_{10} = 0.2$	0.001	(0.049)	95	0.001	(0.049)	96	0.002	(0.049)	95	0.006	(0.051)	93	–0.001	(0.048)	96
$β_{11} = - 0.1$	0.003	(0.019)	96	0.003	(0.019)	95	0.002	(0.019)	95	0.003	(0.021)	93	–0.001	(0.019)	94
$β_{12} = 0.1$	0	(0.04)	95	0	(0.04)	94	–0.001	(0.04)	94	–0.004	(0.041)	93	0.004	(0.039)	97
$β_{13} = - 0.2$	–0.001	(0.034)	97	0	(0.034)	98	0	(0.034)	98	–0.003	(0.035)	96	–0.007	(0.033)	98
$σ_{ε 1} = 0.4$	0	(0.003)	94	0	(0.003)	94	0	(0.003)	94	0.002	(0.01)	92	0.001	(0.004)	95
$β_{20} = 3$	–0.002	(0.053)	96	–0.002	(0.053)	95	0.002	(0.053)	96	0.001	(0.049)	98	–0.008	(0.047)	98
$β_{21} = - 0.1$	–0.002	(0.02)	96	–0.002	(0.02)	96	–0.001	(0.02)	96	–0.002	(0.019)	97	–0.003	(0.019)	98
$β_{22} = 0.1$	0.002	(0.043)	96	0.002	(0.043)	95	0	(0.043)	96	0.001	(0.041)	95	0.005	(0.041)	95
$β_{23} = - 0.2$	0.004	(0.045)	94	0.004	(0.045)	94	0.003	(0.045)	95	0.002	(0.045)	94	0.009	(0.04)	96
$β_{30} = 1$	–0.014	(0.092)	95	–0.014	(0.091)	95	–0.005	(0.093)	94	0.001	(0.096)	94	0.026	(0.102)	97
$β_{31} = - 1$	0.008	(0.031)	93	0.009	(0.031)	93	–0.001	(0.031)	95	–0.001	(0.032)	94	–0.007	(0.031)	96
$β_{32} = 1$	–0.006	(0.074)	96	–0.007	(0.073)	95	0.004	(0.075)	97	–0.002	(0.076)	94	–0.008	(0.071)	98
$β_{33} = - 1$	0.01	(0.07)	95	0.011	(0.071)	95	–0.001	(0.071)	95	–0.003	(0.068)	96	–0.018	(0.077)	85
$σ_{b 10}^{2} = 0.16$	0.003	(0.016)	95	0.002	(0.014)	94	–0.001	(0.014)	99	–0.007	(0.029)	94	–0.008	(0.021)	96
$σ_{b 11}^{2} = 0.09$	0.003	(0.01)	94	0.004	(0.009)	95	0	(0.009)	93	0.008	(0.038)	88	0.011	(0.025)	94
$σ_{b 20}^{2} = 0.25$	0.002	(0.019)	95	0.001	(0.017)	96	0.002	(0.017)	95	0.004	(0.017)	96	0.002	(0.016)	95
$σ_{b 21}^{2} = 0.16$	0.002	(0.014)	95	0.002	(0.013)	95	0.003	(0.014)	94	0.002	(0.012)	97	0.007	(0.016)	82
$σ_{b 30}^{2} = 0.25$	–0.01	(0.036)	96	–0.009	(0.037)	96	0	(0.041)	93	0.004	(0.041)	96	0.002	(0.037)	97
${cov}_{b 10, b 11} = 0.03$	–0.002	(0.009)	96	–0.004	(0.009)	94	–0.001	(0.007)	97	–0.001	(0.01)	96	–0.003	(0.009)	96
${cov}_{b 10, b 20} = 0.02$	0	(0.011)	95	0	(0.011)	95	0	(0.01)	96	–0.001	(0.01)	94	–0.003	(0.011)	97
${cov}_{b 10, b 21} = 0.04$	–0.001	(0.01)	96	–0.001	(0.01)	96	0	(0.009)	96	–0.002	(0.011)	94	0	(0.009)	97
${cov}_{b 10, b 30} = 0$	0	(0.019)	96	–0.003	(0.017)	97	0.001	(0.016)	95	–0.002	(0.016)	94	–0.001	(0.017)	95
${cov}_{b 11, b 20} = 0.03$	–0.001	(0.01)	96	–0.001	(0.01)	96	–0.001	(0.009)	96	0	(0.011)	94	–0.002	(0.01)	97
${cov}_{b 11, b 21} = 0$	0	(0.008)	95	0	(0.008)	96	0	(0.008)	95	0.002	(0.01)	92	0	(0.008)	96
${cov}_{b 11, b 30} = - - 0.06$	0.005	(0.014)	92	0.006	(0.013)	94	0.002	(0.014)	93	0	(0.015)	95	–0.001	(0.014)	97
${cov}_{b 20, b 21} = 0.08$	0	(0.012)	95	0	(0.012)	95	–0.001	(0.011)	94	0	(0.011)	96	0.002	(0.011)	96
${cov}_{b 20, b 30} = 0.05$	–0.001	(0.019)	95	–0.002	(0.019)	95	–0.003	(0.018)	94	–0.001	(0.019)	96	0.001	(0.018)	96
${cov}_{b 21, b 30} = 0.04$	–0.001	(0.017)	95	–0.001	(0.017)	96	–0.003	(0.016)	94	0.009	(0.019)	92	0.011	(0.018)	95
$φ_{1} = 0.5$	–0.003	(0.101)	95	–0.003	(0.102)	94	0	(0.103)	98	0.009	(0.113)	94	0.021	(0.114)	96
$φ_{2} = - 0.2$	0.006	(0.074)	93	0.005	(0.073)	94	0.005	(0.074)	97	0.002	(0.077)	93	0.006	(0.072)	94
$φ_{3} = 0.3$	–0.001	(0.094)	94	0.001	(0.092)	94	–0.004	(0.092)	94	–0.006	(0.093)	94	–0.003	(0.087)	95
Conv. rate	1	1	1	0.51	0.64
Comp. time (s)	43.55 (5.58)	69.8 (44.59)	295.73 (20.22)	3766.41 (1429.06)	10,861.22 (3735.02)

Approach:

R-INLA 1

R-INLA 2

JMbayes2

rstanarm 1

rstanarm 2

(Empirical Bayes)

(Full Bayesian)

(1 chain / 1000 iter.)

(4 chains / 2000 iter.)

True value

Bias

(SD)

CP (%)

Bias

(SD)

Bias

(SD)

Bias

(SD)

Bias

(SD)

β_{10} = 0.2

0.001

(0.049)

0.001

(0.049)

0.002

(0.049)

0.006

(0.051)

–0.001

(0.048)

β_{11} = - 0.1

0.003

(0.019)

0.003

(0.019)

0.002

(0.019)

0.003

(0.021)

–0.001

(0.019)

β_{12} = 0.1

(0.04)

–0.001

(0.04)

–0.004

(0.041)

0.004

(0.039)

β_{13} = - 0.2

–0.001

(0.034)

–0.003

(0.035)

–0.007

(0.033)

σ_{ε 1} = 0.4

(0.003)

0.002

(0.01)

0.001

(0.004)

β_{20} = 3

–0.002

(0.053)

–0.002

(0.053)

0.002

(0.053)

0.001

(0.049)

–0.008

(0.047)

β_{21} = - 0.1

–0.002

(0.02)

–0.002

(0.02)

–0.001

(0.02)

–0.002

(0.019)

–0.003

(0.019)

β_{22} = 0.1

0.002

(0.043)

0.002

(0.043)

0.001

(0.041)

0.005

(0.041)

β_{23} = - 0.2

0.004

(0.045)

0.004

(0.045)

0.003

(0.045)

0.002

(0.045)

0.009

(0.04)

β_{30} = 1

–0.014

(0.092)

–0.014

(0.091)

–0.005

(0.093)

0.001

(0.096)

0.026

(0.102)

β_{31} = - 1

0.008

(0.031)

0.009

(0.031)

–0.001

(0.031)

–0.001

(0.032)

–0.007

(0.031)

β_{32} = 1

–0.006

(0.074)

–0.007

(0.073)

0.004

(0.075)

–0.002

(0.076)

–0.008

(0.071)

β_{33} = - 1

0.01

(0.07)

0.011

(0.071)

–0.001

(0.071)

–0.003

(0.068)

–0.018

(0.077)

σ_{b 10}^{2} = 0.16

0.003

(0.016)

0.002

(0.014)

–0.001

(0.014)

–0.007

(0.029)

–0.008

(0.021)

σ_{b 11}^{2} = 0.09

0.003

(0.01)

0.004

(0.009)

0.008

(0.038)

0.011

(0.025)

σ_{b 20}^{2} = 0.25

0.002

(0.019)

0.001

(0.017)

0.002

(0.017)

0.004

(0.017)

0.002

(0.016)

σ_{b 21}^{2} = 0.16

0.002

(0.014)

0.002

(0.013)

0.003

(0.014)

0.002

(0.012)

0.007

(0.016)

σ_{b 30}^{2} = 0.25

–0.01

(0.036)

–0.009

(0.037)

(0.041)

0.004

(0.041)

0.002

(0.037)

{cov}_{b 10, b 11} = 0.03

–0.002

(0.009)

–0.004

(0.009)

–0.001

(0.007)

–0.001

(0.01)

–0.003

(0.009)

{cov}_{b 10, b 20} = 0.02

(0.011)

(0.01)

–0.001

(0.01)

–0.003

(0.011)

{cov}_{b 10, b 21} = 0.04

–0.001

(0.01)

–0.001

(0.01)

(0.009)

–0.002

(0.011)

(0.009)

{cov}_{b 10, b 30} = 0

(0.019)

–0.003

(0.017)

0.001

(0.016)

–0.002

(0.016)

–0.001

(0.017)

{cov}_{b 11, b 20} = 0.03

–0.001

(0.01)

–0.001

(0.01)

–0.001

(0.009)

(0.011)

–0.002

(0.01)

{cov}_{b 11, b 21} = 0

(0.008)

0.002

(0.01)

(0.008)

{cov}_{b 11, b 30} = - - 0.06

0.005

(0.014)

0.006

(0.013)

0.002

(0.014)

(0.015)

–0.001

(0.014)

{cov}_{b 20, b 21} = 0.08

(0.012)

–0.001

(0.011)

0.002

(0.011)

{cov}_{b 20, b 30} = 0.05

–0.001

(0.019)

–0.002

(0.019)

–0.003

(0.018)

–0.001

(0.019)

0.001

(0.018)

{cov}_{b 21, b 30} = 0.04

–0.001

(0.017)

–0.001

(0.017)

–0.003

(0.016)

0.009

(0.019)

0.011

(0.018)

φ_{1} = 0.5

–0.003

(0.101)

–0.003

(0.102)

(0.103)

0.009

(0.113)

0.021

(0.114)

φ_{2} = - 0.2

0.006

(0.074)

0.005

(0.073)

0.005

(0.074)

0.002

(0.077)

0.006

(0.072)

φ_{3} = 0.3

–0.001

(0.094)

0.001

(0.092)

–0.004

(0.092)

–0.006

(0.093)

–0.003

(0.087)

Conv. rate

0.51

0.64

Comp. time (s)

43.55 (5.58)

69.8 (44.59)

295.73 (20.22)

3766.41 (1429.06)

10,861.22 (3735.02)

Table 2

Open in new tab

Simulations with K=3 longitudinal markers with a mixture of distributions (continuous, count, and binary)

Approach:	R-INLA 1			R-INLA 2			JMbayes2			rstanarm 1			rstanarm 2
	(Empirical Bayes)			(Full Bayesian)						(1 chain / 1000 iter.)			(4 chains / 2000 iter.)
True value	Bias	(SD)	CP (%)	Bias	(SD)	CP	Bias	(SD)	CP	Bias	(SD)	CP	Bias	(SD)	CP
$β_{10} = 0.2$	0.001	(0.049)	95	0.001	(0.049)	96	0.002	(0.049)	95	0.006	(0.051)	93	–0.001	(0.048)	96
$β_{11} = - 0.1$	0.003	(0.019)	96	0.003	(0.019)	95	0.002	(0.019)	95	0.003	(0.021)	93	–0.001	(0.019)	94
$β_{12} = 0.1$	0	(0.04)	95	0	(0.04)	94	–0.001	(0.04)	94	–0.004	(0.041)	93	0.004	(0.039)	97
$β_{13} = - 0.2$	–0.001	(0.034)	97	0	(0.034)	98	0	(0.034)	98	–0.003	(0.035)	96	–0.007	(0.033)	98
$σ_{ε 1} = 0.4$	0	(0.003)	94	0	(0.003)	94	0	(0.003)	94	0.002	(0.01)	92	0.001	(0.004)	95
$β_{20} = 3$	–0.002	(0.053)	96	–0.002	(0.053)	95	0.002	(0.053)	96	0.001	(0.049)	98	–0.008	(0.047)	98
$β_{21} = - 0.1$	–0.002	(0.02)	96	–0.002	(0.02)	96	–0.001	(0.02)	96	–0.002	(0.019)	97	–0.003	(0.019)	98
$β_{22} = 0.1$	0.002	(0.043)	96	0.002	(0.043)	95	0	(0.043)	96	0.001	(0.041)	95	0.005	(0.041)	95
$β_{23} = - 0.2$	0.004	(0.045)	94	0.004	(0.045)	94	0.003	(0.045)	95	0.002	(0.045)	94	0.009	(0.04)	96
$β_{30} = 1$	–0.014	(0.092)	95	–0.014	(0.091)	95	–0.005	(0.093)	94	0.001	(0.096)	94	0.026	(0.102)	97
$β_{31} = - 1$	0.008	(0.031)	93	0.009	(0.031)	93	–0.001	(0.031)	95	–0.001	(0.032)	94	–0.007	(0.031)	96
$β_{32} = 1$	–0.006	(0.074)	96	–0.007	(0.073)	95	0.004	(0.075)	97	–0.002	(0.076)	94	–0.008	(0.071)	98
$β_{33} = - 1$	0.01	(0.07)	95	0.011	(0.071)	95	–0.001	(0.071)	95	–0.003	(0.068)	96	–0.018	(0.077)	85
$σ_{b 10}^{2} = 0.16$	0.003	(0.016)	95	0.002	(0.014)	94	–0.001	(0.014)	99	–0.007	(0.029)	94	–0.008	(0.021)	96
$σ_{b 11}^{2} = 0.09$	0.003	(0.01)	94	0.004	(0.009)	95	0	(0.009)	93	0.008	(0.038)	88	0.011	(0.025)	94
$σ_{b 20}^{2} = 0.25$	0.002	(0.019)	95	0.001	(0.017)	96	0.002	(0.017)	95	0.004	(0.017)	96	0.002	(0.016)	95
$σ_{b 21}^{2} = 0.16$	0.002	(0.014)	95	0.002	(0.013)	95	0.003	(0.014)	94	0.002	(0.012)	97	0.007	(0.016)	82
$σ_{b 30}^{2} = 0.25$	–0.01	(0.036)	96	–0.009	(0.037)	96	0	(0.041)	93	0.004	(0.041)	96	0.002	(0.037)	97
${cov}_{b 10, b 11} = 0.03$	–0.002	(0.009)	96	–0.004	(0.009)	94	–0.001	(0.007)	97	–0.001	(0.01)	96	–0.003	(0.009)	96
${cov}_{b 10, b 20} = 0.02$	0	(0.011)	95	0	(0.011)	95	0	(0.01)	96	–0.001	(0.01)	94	–0.003	(0.011)	97
${cov}_{b 10, b 21} = 0.04$	–0.001	(0.01)	96	–0.001	(0.01)	96	0	(0.009)	96	–0.002	(0.011)	94	0	(0.009)	97
${cov}_{b 10, b 30} = 0$	0	(0.019)	96	–0.003	(0.017)	97	0.001	(0.016)	95	–0.002	(0.016)	94	–0.001	(0.017)	95
${cov}_{b 11, b 20} = 0.03$	–0.001	(0.01)	96	–0.001	(0.01)	96	–0.001	(0.009)	96	0	(0.011)	94	–0.002	(0.01)	97
${cov}_{b 11, b 21} = 0$	0	(0.008)	95	0	(0.008)	96	0	(0.008)	95	0.002	(0.01)	92	0	(0.008)	96
${cov}_{b 11, b 30} = - - 0.06$	0.005	(0.014)	92	0.006	(0.013)	94	0.002	(0.014)	93	0	(0.015)	95	–0.001	(0.014)	97
${cov}_{b 20, b 21} = 0.08$	0	(0.012)	95	0	(0.012)	95	–0.001	(0.011)	94	0	(0.011)	96	0.002	(0.011)	96
${cov}_{b 20, b 30} = 0.05$	–0.001	(0.019)	95	–0.002	(0.019)	95	–0.003	(0.018)	94	–0.001	(0.019)	96	0.001	(0.018)	96
${cov}_{b 21, b 30} = 0.04$	–0.001	(0.017)	95	–0.001	(0.017)	96	–0.003	(0.016)	94	0.009	(0.019)	92	0.011	(0.018)	95
$φ_{1} = 0.5$	–0.003	(0.101)	95	–0.003	(0.102)	94	0	(0.103)	98	0.009	(0.113)	94	0.021	(0.114)	96
$φ_{2} = - 0.2$	0.006	(0.074)	93	0.005	(0.073)	94	0.005	(0.074)	97	0.002	(0.077)	93	0.006	(0.072)	94
$φ_{3} = 0.3$	–0.001	(0.094)	94	0.001	(0.092)	94	–0.004	(0.092)	94	–0.006	(0.093)	94	–0.003	(0.087)	95
Conv. rate	1			1			1			0.51			0.64
Comp. time (s)	43.55 (5.58)			69.8 (44.59)			295.73 (20.22)			3766.41 (1429.06)			10,861.22 (3735.02)

Approach:	R-INLA 1			R-INLA 2			JMbayes2			rstanarm 1			rstanarm 2
	(Empirical Bayes)			(Full Bayesian)						(1 chain / 1000 iter.)			(4 chains / 2000 iter.)
True value	Bias	(SD)	CP (%)	Bias	(SD)	CP	Bias	(SD)	CP	Bias	(SD)	CP	Bias	(SD)	CP
$β_{10} = 0.2$	0.001	(0.049)	95	0.001	(0.049)	96	0.002	(0.049)	95	0.006	(0.051)	93	–0.001	(0.048)	96
$β_{11} = - 0.1$	0.003	(0.019)	96	0.003	(0.019)	95	0.002	(0.019)	95	0.003	(0.021)	93	–0.001	(0.019)	94
$β_{12} = 0.1$	0	(0.04)	95	0	(0.04)	94	–0.001	(0.04)	94	–0.004	(0.041)	93	0.004	(0.039)	97
$β_{13} = - 0.2$	–0.001	(0.034)	97	0	(0.034)	98	0	(0.034)	98	–0.003	(0.035)	96	–0.007	(0.033)	98
$σ_{ε 1} = 0.4$	0	(0.003)	94	0	(0.003)	94	0	(0.003)	94	0.002	(0.01)	92	0.001	(0.004)	95
$β_{20} = 3$	–0.002	(0.053)	96	–0.002	(0.053)	95	0.002	(0.053)	96	0.001	(0.049)	98	–0.008	(0.047)	98
$β_{21} = - 0.1$	–0.002	(0.02)	96	–0.002	(0.02)	96	–0.001	(0.02)	96	–0.002	(0.019)	97	–0.003	(0.019)	98
$β_{22} = 0.1$	0.002	(0.043)	96	0.002	(0.043)	95	0	(0.043)	96	0.001	(0.041)	95	0.005	(0.041)	95
$β_{23} = - 0.2$	0.004	(0.045)	94	0.004	(0.045)	94	0.003	(0.045)	95	0.002	(0.045)	94	0.009	(0.04)	96
$β_{30} = 1$	–0.014	(0.092)	95	–0.014	(0.091)	95	–0.005	(0.093)	94	0.001	(0.096)	94	0.026	(0.102)	97
$β_{31} = - 1$	0.008	(0.031)	93	0.009	(0.031)	93	–0.001	(0.031)	95	–0.001	(0.032)	94	–0.007	(0.031)	96
$β_{32} = 1$	–0.006	(0.074)	96	–0.007	(0.073)	95	0.004	(0.075)	97	–0.002	(0.076)	94	–0.008	(0.071)	98
$β_{33} = - 1$	0.01	(0.07)	95	0.011	(0.071)	95	–0.001	(0.071)	95	–0.003	(0.068)	96	–0.018	(0.077)	85
$σ_{b 10}^{2} = 0.16$	0.003	(0.016)	95	0.002	(0.014)	94	–0.001	(0.014)	99	–0.007	(0.029)	94	–0.008	(0.021)	96
$σ_{b 11}^{2} = 0.09$	0.003	(0.01)	94	0.004	(0.009)	95	0	(0.009)	93	0.008	(0.038)	88	0.011	(0.025)	94
$σ_{b 20}^{2} = 0.25$	0.002	(0.019)	95	0.001	(0.017)	96	0.002	(0.017)	95	0.004	(0.017)	96	0.002	(0.016)	95
$σ_{b 21}^{2} = 0.16$	0.002	(0.014)	95	0.002	(0.013)	95	0.003	(0.014)	94	0.002	(0.012)	97	0.007	(0.016)	82
$σ_{b 30}^{2} = 0.25$	–0.01	(0.036)	96	–0.009	(0.037)	96	0	(0.041)	93	0.004	(0.041)	96	0.002	(0.037)	97
${cov}_{b 10, b 11} = 0.03$	–0.002	(0.009)	96	–0.004	(0.009)	94	–0.001	(0.007)	97	–0.001	(0.01)	96	–0.003	(0.009)	96
${cov}_{b 10, b 20} = 0.02$	0	(0.011)	95	0	(0.011)	95	0	(0.01)	96	–0.001	(0.01)	94	–0.003	(0.011)	97
${cov}_{b 10, b 21} = 0.04$	–0.001	(0.01)	96	–0.001	(0.01)	96	0	(0.009)	96	–0.002	(0.011)	94	0	(0.009)	97
${cov}_{b 10, b 30} = 0$	0	(0.019)	96	–0.003	(0.017)	97	0.001	(0.016)	95	–0.002	(0.016)	94	–0.001	(0.017)	95
${cov}_{b 11, b 20} = 0.03$	–0.001	(0.01)	96	–0.001	(0.01)	96	–0.001	(0.009)	96	0	(0.011)	94	–0.002	(0.01)	97
${cov}_{b 11, b 21} = 0$	0	(0.008)	95	0	(0.008)	96	0	(0.008)	95	0.002	(0.01)	92	0	(0.008)	96
${cov}_{b 11, b 30} = - - 0.06$	0.005	(0.014)	92	0.006	(0.013)	94	0.002	(0.014)	93	0	(0.015)	95	–0.001	(0.014)	97
${cov}_{b 20, b 21} = 0.08$	0	(0.012)	95	0	(0.012)	95	–0.001	(0.011)	94	0	(0.011)	96	0.002	(0.011)	96
${cov}_{b 20, b 30} = 0.05$	–0.001	(0.019)	95	–0.002	(0.019)	95	–0.003	(0.018)	94	–0.001	(0.019)	96	0.001	(0.018)	96
${cov}_{b 21, b 30} = 0.04$	–0.001	(0.017)	95	–0.001	(0.017)	96	–0.003	(0.016)	94	0.009	(0.019)	92	0.011	(0.018)	95
$φ_{1} = 0.5$	–0.003	(0.101)	95	–0.003	(0.102)	94	0	(0.103)	98	0.009	(0.113)	94	0.021	(0.114)	96
$φ_{2} = - 0.2$	0.006	(0.074)	93	0.005	(0.073)	94	0.005	(0.074)	97	0.002	(0.077)	93	0.006	(0.072)	94
$φ_{3} = 0.3$	–0.001	(0.094)	94	0.001	(0.092)	94	–0.004	(0.092)	94	–0.006	(0.093)	94	–0.003	(0.087)	95
Conv. rate	1			1			1			0.51			0.64
Comp. time (s)	43.55 (5.58)			69.8 (44.59)			295.73 (20.22)			3766.41 (1429.06)			10,861.22 (3735.02)

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